Exponential Decay of Quasilinear Maxwell Equations with Interior Conductivity

TL;DR

This paper proves that solutions to a quasilinear Maxwell system with interior conductivity decay exponentially over time, under small initial data, by establishing energy estimates and overcoming technical challenges.

Contribution

It introduces a novel energy barrier estimate for quasilinear Maxwell equations with anisotropy and conductivity, leading to global existence and exponential decay results.

Findings

Solutions decay exponentially over time.

Global existence of solutions under small initial data.

Addresses technical challenges of quasilinearity and anisotropy.

Abstract

We consider a quasilinear nonhomogeneous, anisotropic Maxwell system in a bounded smooth domain of $\mathbb{R}^{3}$ with a strictly positive conductivity subject to the boundary conditions of a perfect conductor. Under appropriate regularity conditions, adopting a classical $L^{2}$-Sobolev solution framework, a nonlinear energy barrier estimate is established for local-in-time $H^{3}$-solutions to the Maxwell system by a proper combination of higher-order energy and observability-type estimates under a smallness assumption on the initial data. Technical complications due to quasilinearity, anisotropy and the lack of solenoidality, etc., are addressed. Finally, provided the initial data are small, the barrier method is applied to prove that local solutions exist globally and exhibit an exponential decay rate.